\(\int \frac {(f+g x)^2 (A+B x+C x^2)}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx\) [57]

Optimal result

Integrand size = 41, antiderivative size = 1068 \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx =\text {Too large to display} \] Output:

2/105*(24*b^2*C*e^2*g^2-c*e*g*(25*a*C*e*g+b*(28*B*e*g-41*C*d*g+56*C*e*f))+ 
c^2*(7*e*g*(5*A*e*g-7*B*d*g+10*B*e*f)+C*(42*d^2*g^2-68*d*e*f*g+20*e^2*f^2) 
))*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^3/e^3-2/35*g*(6*b*C*e*g-c*(7*B*e*g- 
6*C*d*g+4*C*e*f))*(e*x+d)^(3/2)*(c*x^2+b*x+a)^(1/2)/c^2/e^3+2/7*C*(e*x+d)^ 
(1/2)*(g*x+f)^2*(c*x^2+b*x+a)^(1/2)/c/e-1/105*2^(1/2)*(-4*a*c+b^2)^(1/2)*( 
48*b^3*C*e^3*g^2-8*b*c*e^2*g*(13*a*C*e*g+b*(7*B*e*g-5*C*d*g+14*C*e*f))+c^2 
*e*(7*b*e*g*(10*A*e*g-7*B*d*g+20*B*e*f)+a*e*g*(63*B*e*g-44*C*d*g+126*C*e*f 
)+b*C*(40*d^2*g^2-98*d*e*f*g+70*e^2*f^2))+c^3*(2*C*d*(24*d^2*g^2-56*d*e*f* 
g+35*e^2*f^2)-7*e*(10*A*e*g*(-d*g+3*e*f)+B*(8*d^2*g^2-20*d*e*f*g+15*e^2*f^ 
2))))*(e*x+d)^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*EllipticE(1/2*(1 
+(2*c*x+b)/(-4*a*c+b^2)^(1/2))^(1/2)*2^(1/2),(-2*(-4*a*c+b^2)^(1/2)*e/(2*c 
*d-(b+(-4*a*c+b^2)^(1/2))*e))^(1/2))/c^4/e^4/(c*(e*x+d)/(2*c*d-(b+(-4*a*c+ 
b^2)^(1/2))*e))^(1/2)/(c*x^2+b*x+a)^(1/2)+2/105*2^(1/2)*(-4*a*c+b^2)^(1/2) 
*(24*b^2*C*e^3*(-a*e+b*d)*g^2+c*e^2*g*(25*a^2*C*e^2*g+2*a*b*e*(14*B*e*g-33 
*C*d*g+28*C*e*f)-b^2*d*(28*B*e*g-17*C*d*g+56*C*e*f))+c^3*(2*C*d^2*(24*d^2* 
g^2-56*d*e*f*g+35*e^2*f^2)+7*e*(5*A*e*(2*d^2*g^2-6*d*e*f*g+3*e^2*f^2)-B*d* 
(8*d^2*g^2-20*d*e*f*g+15*e^2*f^2)))+c^2*e*(b*d*(7*e*g*(5*A*e*g-3*B*d*g+10* 
B*e*f)+C*(16*d^2*g^2-42*d*e*f*g+35*e^2*f^2))-a*e*(7*e*g*(5*A*e*g-7*B*d*g+1 
0*B*e*f)+C*(32*d^2*g^2-98*d*e*f*g+35*e^2*f^2))))*(c*(e*x+d)/(2*c*d-(b+(-4* 
a*c+b^2)^(1/2))*e))^(1/2)*(-c*(c*x^2+b*x+a)/(-4*a*c+b^2))^(1/2)*Ellipti...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 38.92 (sec) , antiderivative size = 37177, normalized size of antiderivative = 34.81 \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\text {Result too large to show} \] Input:

Integrate[((f + g*x)^2*(A + B*x + C*x^2))/(Sqrt[d + e*x]*Sqrt[a + b*x + c* 
x^2]),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 5.47 (sec) , antiderivative size = 1109, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2184, 27, 2184, 27, 2184, 27, 1269, 1172, 321, 327}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {2 \int \frac {e^3 g (14 c C e f-16 c C d g+7 B c e g-6 b C e g) x^3-e^2 \left (-7 c g (2 B f+A g) e^2+C (13 b d+5 a e) g^2 e-c C \left (7 e^2 f^2-11 d^2 g^2\right )\right ) x^2+e \left (7 B c f^2 e^3+2 g \left (7 A c e^3 f-C d \left (c d^2+4 b e d+5 a e^2\right ) g\right )\right ) x+e \left (7 A c e^3 f^2-C d^2 (b d+5 a e) g^2\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{7 c e^4}+\frac {2 C g^2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {e^3 g (14 c C e f-16 c C d g+7 B c e g-6 b C e g) x^3-e^2 \left (-7 c g (2 B f+A g) e^2+C (13 b d+5 a e) g^2 e-c C \left (7 e^2 f^2-11 d^2 g^2\right )\right ) x^2+e \left (7 B c f^2 e^3+2 g \left (7 A c e^3 f-C d \left (c d^2+4 b e d+5 a e^2\right ) g\right )\right ) x+e \left (7 A c e^3 f^2-C d^2 (b d+5 a e) g^2\right )}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{7 c e^4}+\frac {2 C g^2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^3}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {\frac {2 \int \frac {\left (\left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+57 d^2 g^2\right )\right ) c^2-e g (25 a C e g+b (56 C e f-41 C d g+28 B e g)) c+24 b^2 C e^2 g^2\right ) x^2 e^5+\left (35 A c^2 f^2 e^3+d g \left (6 C d e g b^2+18 a C e^2 g b-c d (14 C e f-11 C d g+7 B e g) b-a c e (42 C e f-23 C d g+21 B e g)\right )\right ) e^4-\left (7 B c e \left (e (5 b d+3 a e) g^2-c \left (5 e^2 f^2-2 d^2 g^2\right )\right )-2 g \left (\left (35 A e^3 f-C d^2 (14 e f-11 d g)\right ) c^2-C e (b d (35 e f-26 d g)+a e (21 e f+d g)) c+3 b C e^2 (5 b d+3 a e) g\right )\right ) x e^4}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{5 c e^3}+\frac {2 e g (d+e x)^{3/2} \sqrt {a+b x+c x^2} (-6 b C e g+7 B c e g-16 c C d g+14 c C e f)}{5 c}}{7 c e^4}+\frac {2 C g^2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\int \frac {\left (\left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+57 d^2 g^2\right )\right ) c^2-e g (25 a C e g+b (56 C e f-41 C d g+28 B e g)) c+24 b^2 C e^2 g^2\right ) x^2 e^5+\left (35 A c^2 f^2 e^3+d g \left (6 C d e g b^2+18 a C e^2 g b-c d (14 C e f-11 C d g+7 B e g) b-a c e (42 C e f-23 C d g+21 B e g)\right )\right ) e^4-\left (7 B c e \left (e (5 b d+3 a e) g^2-c \left (5 e^2 f^2-2 d^2 g^2\right )\right )-2 g \left (\left (35 A e^3 f-C d^2 (14 e f-11 d g)\right ) c^2-C e (b d (35 e f-26 d g)+a e (21 e f+d g)) c+3 b C e^2 (5 b d+3 a e) g\right )\right ) x e^4}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{5 c e^3}+\frac {2 e g (d+e x)^{3/2} \sqrt {a+b x+c x^2} (-6 b C e g+7 B c e g-16 c C d g+14 c C e f)}{5 c}}{7 c e^4}+\frac {2 C g^2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^3}\)

\(\Big \downarrow \) 2184

\(\displaystyle \frac {\frac {\frac {2 \int -\frac {e^6 \left (24 C d e^2 g^2 b^3+e g \left (24 a C e^2 g-c d (56 C e f-23 C d g+28 B e g)\right ) b^2-c \left (2 a e^2 g (28 C e f+19 C d g+14 B e g)-c d \left (7 e g (10 B e f-4 B d g+5 A e g)+C \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right )\right )\right ) b-c e \left (35 A c \left (3 c f^2-a g^2\right ) e^2+a \left (25 a C e^2 g^2-14 B c e (5 e f+d g) g-c C \left (35 e^2 f^2+28 d e g f-12 d^2 g^2\right )\right )\right )+\left (\left (2 C d \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right )-7 e \left (10 A e g (3 e f-d g)+B \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (7 b e g (20 B e f-7 B d g+10 A e g)+a e g (126 C e f-44 C d g+63 B e g)+b C \left (70 e^2 f^2-98 d e g f+40 d^2 g^2\right )\right ) c^2-8 b e^2 g (13 a C e g+b (14 C e f-5 C d g+7 B e g)) c+48 b^3 C e^3 g^2\right ) x\right )}{2 \sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c e^2}+\frac {2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e g (25 a C e g+b (28 B e g-41 C d g+56 C e f))+c^2 \left (7 e g (5 A e g-7 B d g+10 B e f)+C \left (57 d^2 g^2-98 d e f g+35 e^2 f^2\right )\right )+24 b^2 C e^2 g^2\right )}{3 c}}{5 c e^3}+\frac {2 e g (d+e x)^{3/2} \sqrt {a+b x+c x^2} (-6 b C e g+7 B c e g-16 c C d g+14 c C e f)}{5 c}}{7 c e^4}+\frac {2 C g^2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e g (25 a C e g+b (28 B e g-41 C d g+56 C e f))+c^2 \left (7 e g (5 A e g-7 B d g+10 B e f)+C \left (57 d^2 g^2-98 d e f g+35 e^2 f^2\right )\right )+24 b^2 C e^2 g^2\right )}{3 c}-\frac {e^4 \int \frac {24 C d e^2 g^2 b^3+e g \left (24 a C e^2 g-c d (56 C e f-23 C d g+28 B e g)\right ) b^2-c \left (2 a e^2 g (28 C e f+19 C d g+14 B e g)-c d \left (7 e g (10 B e f-4 B d g+5 A e g)+C \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right )\right )\right ) b-c e \left (35 A c \left (3 c f^2-a g^2\right ) e^2+a \left (25 a C e^2 g^2-14 B c e (5 e f+d g) g-c C \left (35 e^2 f^2+28 d e g f-12 d^2 g^2\right )\right )\right )+\left (\left (2 C d \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right )-7 e \left (10 A e g (3 e f-d g)+B \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (7 b e g (20 B e f-7 B d g+10 A e g)+a e g (126 C e f-44 C d g+63 B e g)+b C \left (70 e^2 f^2-98 d e g f+40 d^2 g^2\right )\right ) c^2-8 b e^2 g (13 a C e g+b (14 C e f-5 C d g+7 B e g)) c+48 b^3 C e^3 g^2\right ) x}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{3 c}}{5 c e^3}+\frac {2 e g (d+e x)^{3/2} \sqrt {a+b x+c x^2} (-6 b C e g+7 B c e g-16 c C d g+14 c C e f)}{5 c}}{7 c e^4}+\frac {2 C g^2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^3}\)

\(\Big \downarrow \) 1269

\(\displaystyle \frac {\frac {\frac {2 e^4 \sqrt {d+e x} \sqrt {a+b x+c x^2} \left (-c e g (25 a C e g+b (28 B e g-41 C d g+56 C e f))+c^2 \left (7 e g (5 A e g-7 B d g+10 B e f)+C \left (57 d^2 g^2-98 d e f g+35 e^2 f^2\right )\right )+24 b^2 C e^2 g^2\right )}{3 c}-\frac {e^4 \left (\frac {\left (c^2 e \left (a e g (63 B e g-44 C d g+126 C e f)+7 b e g (10 A e g-7 B d g+20 B e f)+b C \left (40 d^2 g^2-98 d e f g+70 e^2 f^2\right )\right )-8 b c e^2 g (13 a C e g+b (7 B e g-5 C d g+14 C e f))+c^3 \left (2 C d \left (24 d^2 g^2-56 d e f g+35 e^2 f^2\right )-7 e \left (10 A e g (3 e f-d g)+B \left (8 d^2 g^2-20 d e f g+15 e^2 f^2\right )\right )\right )+48 b^3 C e^3 g^2\right ) \int \frac {\sqrt {d+e x}}{\sqrt {c x^2+b x+a}}dx}{e}-\frac {\left (c e^2 g \left (25 a^2 C e^2 g+2 a b e (14 B e g-33 C d g+28 C e f)+b^2 (-d) (28 B e g-17 C d g+56 C e f)\right )+c^2 e \left (b d \left (7 e g (5 A e g-3 B d g+10 B e f)+C \left (16 d^2 g^2-42 d e f g+35 e^2 f^2\right )\right )-a e \left (7 e g (5 A e g-7 B d g+10 B e f)+C \left (32 d^2 g^2-98 d e f g+35 e^2 f^2\right )\right )\right )+24 b^2 C e^3 g^2 (b d-a e)+c^3 \left (7 e \left (5 A e \left (2 d^2 g^2-6 d e f g+3 e^2 f^2\right )-B d \left (8 d^2 g^2-20 d e f g+15 e^2 f^2\right )\right )+2 C d^2 \left (24 d^2 g^2-56 d e f g+35 e^2 f^2\right )\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {c x^2+b x+a}}dx}{e}\right )}{3 c}}{5 c e^3}+\frac {2 e g (d+e x)^{3/2} \sqrt {a+b x+c x^2} (-6 b C e g+7 B c e g-16 c C d g+14 c C e f)}{5 c}}{7 c e^4}+\frac {2 C g^2 (d+e x)^{5/2} \sqrt {a+b x+c x^2}}{7 c e^3}\)

\(\Big \downarrow \) 1172

\(\displaystyle \frac {2 C g^2 \sqrt {c x^2+b x+a} (d+e x)^{5/2}}{7 c e^3}+\frac {\frac {2 e g (14 c C e f-16 c C d g+7 B c e g-6 b C e g) \sqrt {c x^2+b x+a} (d+e x)^{3/2}}{5 c}+\frac {\frac {2 e^4 \left (\left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+57 d^2 g^2\right )\right ) c^2-e g (25 a C e g+b (56 C e f-41 C d g+28 B e g)) c+24 b^2 C e^2 g^2\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e^4 \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C d \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right )-7 e \left (10 A e g (3 e f-d g)+B \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (7 b e g (20 B e f-7 B d g+10 A e g)+a e g (126 C e f-44 C d g+63 B e g)+b C \left (70 e^2 f^2-98 d e g f+40 d^2 g^2\right )\right ) c^2-8 b e^2 g (13 a C e g+b (14 C e f-5 C d g+7 B e g)) c+48 b^3 C e^3 g^2\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right ) d^2+7 e \left (5 A e \left (3 e^2 f^2-6 d e g f+2 d^2 g^2\right )-B d \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (b d \left (7 e g (10 B e f-3 B d g+5 A e g)+C \left (35 e^2 f^2-42 d e g f+16 d^2 g^2\right )\right )-a e \left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+32 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (56 C e f-17 C d g+28 B e g) b^2+2 a e (28 C e f-33 C d g+14 B e g) b+25 a^2 C e^2 g\right ) c+24 b^2 C e^3 (b d-a e) g^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {1}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}} \sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{3 c}}{5 c e^3}}{7 c e^4}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {2 C g^2 \sqrt {c x^2+b x+a} (d+e x)^{5/2}}{7 c e^3}+\frac {\frac {2 e g (14 c C e f-16 c C d g+7 B c e g-6 b C e g) \sqrt {c x^2+b x+a} (d+e x)^{3/2}}{5 c}+\frac {\frac {2 e^4 \left (\left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+57 d^2 g^2\right )\right ) c^2-e g (25 a C e g+b (56 C e f-41 C d g+28 B e g)) c+24 b^2 C e^2 g^2\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e^4 \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C d \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right )-7 e \left (10 A e g (3 e f-d g)+B \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (7 b e g (20 B e f-7 B d g+10 A e g)+a e g (126 C e f-44 C d g+63 B e g)+b C \left (70 e^2 f^2-98 d e g f+40 d^2 g^2\right )\right ) c^2-8 b e^2 g (13 a C e g+b (14 C e f-5 C d g+7 B e g)) c+48 b^3 C e^3 g^2\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \int \frac {\sqrt {\frac {e \left (b+2 c x+\sqrt {b^2-4 a c}\right )}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}+1}}{\sqrt {1-\frac {b+2 c x+\sqrt {b^2-4 a c}}{2 \sqrt {b^2-4 a c}}}}d\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right ) d^2+7 e \left (5 A e \left (3 e^2 f^2-6 d e g f+2 d^2 g^2\right )-B d \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (b d \left (7 e g (10 B e f-3 B d g+5 A e g)+C \left (35 e^2 f^2-42 d e g f+16 d^2 g^2\right )\right )-a e \left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+32 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (56 C e f-17 C d g+28 B e g) b^2+2 a e (28 C e f-33 C d g+14 B e g) b+25 a^2 C e^2 g\right ) c+24 b^2 C e^3 (b d-a e) g^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{3 c}}{5 c e^3}}{7 c e^4}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {2 C g^2 \sqrt {c x^2+b x+a} (d+e x)^{5/2}}{7 c e^3}+\frac {\frac {2 e g (14 c C e f-16 c C d g+7 B c e g-6 b C e g) \sqrt {c x^2+b x+a} (d+e x)^{3/2}}{5 c}+\frac {\frac {2 e^4 \left (\left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+57 d^2 g^2\right )\right ) c^2-e g (25 a C e g+b (56 C e f-41 C d g+28 B e g)) c+24 b^2 C e^2 g^2\right ) \sqrt {d+e x} \sqrt {c x^2+b x+a}}{3 c}-\frac {e^4 \left (\frac {\sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C d \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right )-7 e \left (10 A e g (3 e f-d g)+B \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (7 b e g (20 B e f-7 B d g+10 A e g)+a e g (126 C e f-44 C d g+63 B e g)+b C \left (70 e^2 f^2-98 d e g f+40 d^2 g^2\right )\right ) c^2-8 b e^2 g (13 a C e g+b (14 C e f-5 C d g+7 B e g)) c+48 b^3 C e^3 g^2\right ) \sqrt {d+e x} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} E\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right )|-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {c x^2+b x+a}}-\frac {2 \sqrt {2} \sqrt {b^2-4 a c} \left (\left (2 C \left (35 e^2 f^2-56 d e g f+24 d^2 g^2\right ) d^2+7 e \left (5 A e \left (3 e^2 f^2-6 d e g f+2 d^2 g^2\right )-B d \left (15 e^2 f^2-20 d e g f+8 d^2 g^2\right )\right )\right ) c^3+e \left (b d \left (7 e g (10 B e f-3 B d g+5 A e g)+C \left (35 e^2 f^2-42 d e g f+16 d^2 g^2\right )\right )-a e \left (7 e g (10 B e f-7 B d g+5 A e g)+C \left (35 e^2 f^2-98 d e g f+32 d^2 g^2\right )\right )\right ) c^2+e^2 g \left (-d (56 C e f-17 C d g+28 B e g) b^2+2 a e (28 C e f-33 C d g+14 B e g) b+25 a^2 C e^2 g\right ) c+24 b^2 C e^3 (b d-a e) g^2\right ) \sqrt {\frac {c (d+e x)}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}} \sqrt {-\frac {c \left (c x^2+b x+a\right )}{b^2-4 a c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {b+2 c x+\sqrt {b^2-4 a c}}{\sqrt {b^2-4 a c}}}}{\sqrt {2}}\right ),-\frac {2 \sqrt {b^2-4 a c} e}{2 c d-\left (b+\sqrt {b^2-4 a c}\right ) e}\right )}{c e \sqrt {d+e x} \sqrt {c x^2+b x+a}}\right )}{3 c}}{5 c e^3}}{7 c e^4}\)

Input:

Int[((f + g*x)^2*(A + B*x + C*x^2))/(Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]), 
x]
 

Output:

(2*C*g^2*(d + e*x)^(5/2)*Sqrt[a + b*x + c*x^2])/(7*c*e^3) + ((2*e*g*(14*c* 
C*e*f - 16*c*C*d*g + 7*B*c*e*g - 6*b*C*e*g)*(d + e*x)^(3/2)*Sqrt[a + b*x + 
 c*x^2])/(5*c) + ((2*e^4*(24*b^2*C*e^2*g^2 - c*e*g*(25*a*C*e*g + b*(56*C*e 
*f - 41*C*d*g + 28*B*e*g)) + c^2*(7*e*g*(10*B*e*f - 7*B*d*g + 5*A*e*g) + C 
*(35*e^2*f^2 - 98*d*e*f*g + 57*d^2*g^2)))*Sqrt[d + e*x]*Sqrt[a + b*x + c*x 
^2])/(3*c) - (e^4*((Sqrt[2]*Sqrt[b^2 - 4*a*c]*(48*b^3*C*e^3*g^2 - 8*b*c*e^ 
2*g*(13*a*C*e*g + b*(14*C*e*f - 5*C*d*g + 7*B*e*g)) + c^2*e*(7*b*e*g*(20*B 
*e*f - 7*B*d*g + 10*A*e*g) + a*e*g*(126*C*e*f - 44*C*d*g + 63*B*e*g) + b*C 
*(70*e^2*f^2 - 98*d*e*f*g + 40*d^2*g^2)) + c^3*(2*C*d*(35*e^2*f^2 - 56*d*e 
*f*g + 24*d^2*g^2) - 7*e*(10*A*e*g*(3*e*f - d*g) + B*(15*e^2*f^2 - 20*d*e* 
f*g + 8*d^2*g^2))))*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a* 
c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a* 
c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e) 
])/(c*e*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b 
*x + c*x^2]) - (2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(24*b^2*C*e^3*(b*d - a*e)*g^2 
+ c*e^2*g*(25*a^2*C*e^2*g + 2*a*b*e*(28*C*e*f - 33*C*d*g + 14*B*e*g) - b^2 
*d*(56*C*e*f - 17*C*d*g + 28*B*e*g)) + c^3*(2*C*d^2*(35*e^2*f^2 - 56*d*e*f 
*g + 24*d^2*g^2) + 7*e*(5*A*e*(3*e^2*f^2 - 6*d*e*f*g + 2*d^2*g^2) - B*d*(1 
5*e^2*f^2 - 20*d*e*f*g + 8*d^2*g^2))) + c^2*e*(b*d*(7*e*g*(10*B*e*f - 3*B* 
d*g + 5*A*e*g) + C*(35*e^2*f^2 - 42*d*e*f*g + 16*d^2*g^2)) - a*e*(7*e*g...
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 1172

Int[((d_.) + (e_.)*(x_))^(m_)/Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2], x_Sy 
mbol] :> Simp[2*Rt[b^2 - 4*a*c, 2]*(d + e*x)^m*(Sqrt[(-c)*((a + b*x + c*x^2 
)/(b^2 - 4*a*c))]/(c*Sqrt[a + b*x + c*x^2]*(2*c*((d + e*x)/(2*c*d - b*e - e 
*Rt[b^2 - 4*a*c, 2])))^m))   Subst[Int[(1 + 2*e*Rt[b^2 - 4*a*c, 2]*(x^2/(2* 
c*d - b*e - e*Rt[b^2 - 4*a*c, 2])))^m/Sqrt[1 - x^2], x], x, Sqrt[(b + Rt[b^ 
2 - 4*a*c, 2] + 2*c*x)/(2*Rt[b^2 - 4*a*c, 2])]], x] /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[m^2, 1/4]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2184

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c 
*d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ 
q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol 
yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !(IGt 
Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
 

Maple [A] (verified)

Time = 18.72 (sec) , antiderivative size = 1316, normalized size of antiderivative = 1.23

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1316\)
risch	\(\text {Expression too large to display}\)	\(7433\)
default	\(\text {Expression too large to display}\)	\(25662\)

Input:

int((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x,method=_RE 
TURNVERBOSE)
 

Output:

((e*x+d)*(c*x^2+b*x+a))^(1/2)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2)*(2/7*C*g^2 
/c/e*x^2*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2/5*(B*g^2+2*C*f* 
g-2/7*C*g^2/c/e*(3*b*e+3*c*d))/c/e*x*(c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+ 
a*d)^(1/2)+2/3*(A*g^2+2*B*f*g+C*f^2-2/7*C*g^2/c/e*(5/2*a*e+5/2*b*d)-2/5*(B 
*g^2+2*C*f*g-2/7*C*g^2/c/e*(3*b*e+3*c*d))/c/e*(2*b*e+2*c*d))/c/e*(c*e*x^3+ 
b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)^(1/2)+2*(A*f^2-2/5*(B*g^2+2*C*f*g-2/7*C*g 
^2/c/e*(3*b*e+3*c*d))/c/e*d*a-2/3*(A*g^2+2*B*f*g+C*f^2-2/7*C*g^2/c/e*(5/2* 
a*e+5/2*b*d)-2/5*(B*g^2+2*C*f*g-2/7*C*g^2/c/e*(3*b*e+3*c*d))/c/e*(2*b*e+2* 
c*d))/c/e*(1/2*a*e+1/2*b*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c)*((x+d/e)/( 
d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+b^2)^(1/2)) 
)/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4*a*c+b^2)^(1/2 
))/c)/(-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)/(c*e*x^3+b*e*x^2+c*d*x^2+ 
a*e*x+b*d*x+a*d)^(1/2)*EllipticF(((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/ 
c))^(1/2),((-d/e+1/2*(b+(-4*a*c+b^2)^(1/2))/c)/(-d/e-1/2/c*(-b+(-4*a*c+b^2 
)^(1/2))))^(1/2))+2*(2*A*f*g+B*f^2-4/7*C*g^2/c/e*d*a-2/5*(B*g^2+2*C*f*g-2/ 
7*C*g^2/c/e*(3*b*e+3*c*d))/c/e*(3/2*a*e+3/2*b*d)-2/3*(A*g^2+2*B*f*g+C*f^2- 
2/7*C*g^2/c/e*(5/2*a*e+5/2*b*d)-2/5*(B*g^2+2*C*f*g-2/7*C*g^2/c/e*(3*b*e+3* 
c*d))/c/e*(2*b*e+2*c*d))/c/e*(b*e+c*d))*(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c) 
*((x+d/e)/(d/e-1/2*(b+(-4*a*c+b^2)^(1/2))/c))^(1/2)*((x-1/2/c*(-b+(-4*a*c+ 
b^2)^(1/2)))/(-d/e-1/2/c*(-b+(-4*a*c+b^2)^(1/2))))^(1/2)*((x+1/2*(b+(-4...
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 1121, normalized size of antiderivative = 1.05 \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\text {Too large to display} \] Input:

integrate((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="fricas")
 

Output:

2/315*((35*(2*C*c^4*d^2*e^2 + (C*b*c^3 - 3*B*c^4)*d*e^3 + (2*C*b^2*c^2 + 9 
*A*c^4 - 3*(C*a + B*b)*c^3)*e^4)*f^2 - 14*(8*C*c^4*d^3*e + (3*C*b*c^3 - 10 
*B*c^4)*d^2*e^2 + (3*C*b^2*c^2 + 15*A*c^4 - (3*C*a + 5*B*b)*c^3)*d*e^3 + ( 
8*C*b^3*c + 15*(B*a + A*b)*c^3 - (21*C*a*b + 10*B*b^2)*c^2)*e^4)*f*g + (48 
*C*c^4*d^4 + 8*(2*C*b*c^3 - 7*B*c^4)*d^3*e + (11*C*b^2*c^2 + 70*A*c^4 - (8 
*C*a + 21*B*b)*c^3)*d^2*e^2 + (16*C*b^3*c + 7*(3*B*a + 5*A*b)*c^3 - (34*C* 
a*b + 21*B*b^2)*c^2)*d*e^3 + (48*C*b^4 - 105*A*a*c^3 + (75*C*a^2 + 147*B*a 
*b + 70*A*b^2)*c^2 - 8*(22*C*a*b^2 + 7*B*b^3)*c)*e^4)*g^2)*sqrt(c*e)*weier 
strassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/2 
7*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b* 
c)*e^3)/(c^3*e^3), 1/3*(3*c*e*x + c*d + b*e)/(c*e)) + 3*(35*(2*C*c^4*d*e^3 
 + (2*C*b*c^3 - 3*B*c^4)*e^4)*f^2 - 14*(8*C*c^4*d^2*e^2 + (7*C*b*c^3 - 10* 
B*c^4)*d*e^3 + (8*C*b^2*c^2 + 15*A*c^4 - (9*C*a + 10*B*b)*c^3)*e^4)*f*g + 
(48*C*c^4*d^3*e + 8*(5*C*b*c^3 - 7*B*c^4)*d^2*e^2 + (40*C*b^2*c^2 + 70*A*c 
^4 - (44*C*a + 49*B*b)*c^3)*d*e^3 + (48*C*b^3*c + 7*(9*B*a + 10*A*b)*c^3 - 
 8*(13*C*a*b + 7*B*b^2)*c^2)*e^4)*g^2)*sqrt(c*e)*weierstrassZeta(4/3*(c^2* 
d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/27*(2*c^3*d^3 - 3*b*c^2*d 
^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a*b*c)*e^3)/(c^3*e^3), weier 
strassPInverse(4/3*(c^2*d^2 - b*c*d*e + (b^2 - 3*a*c)*e^2)/(c^2*e^2), -4/2 
7*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*(b^2*c - 6*a*c^2)*d*e^2 + (2*b^3 - 9*a...
 

Sympy [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (f + g x\right )^{2} \left (A + B x + C x^{2}\right )}{\sqrt {d + e x} \sqrt {a + b x + c x^{2}}}\, dx \] Input:

integrate((g*x+f)**2*(C*x**2+B*x+A)/(e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x 
)
 

Output:

Integral((f + g*x)**2*(A + B*x + C*x**2)/(sqrt(d + e*x)*sqrt(a + b*x + c*x 
**2)), x)
 

Maxima [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (g x + f\right )}^{2}}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d}} \,d x } \] Input:

integrate((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(g*x + f)^2/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + 
d)), x)
 

Giac [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (g x + f\right )}^{2}}{\sqrt {c x^{2} + b x + a} \sqrt {e x + d}} \,d x } \] Input:

integrate((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x, alg 
orithm="giac")
 

Output:

integrate((C*x^2 + B*x + A)*(g*x + f)^2/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + 
d)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {{\left (f+g\,x\right )}^2\,\left (C\,x^2+B\,x+A\right )}{\sqrt {d+e\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \] Input:

int(((f + g*x)^2*(A + B*x + C*x^2))/((d + e*x)^(1/2)*(a + b*x + c*x^2)^(1/ 
2)),x)
 

Output:

int(((f + g*x)^2*(A + B*x + C*x^2))/((d + e*x)^(1/2)*(a + b*x + c*x^2)^(1/ 
2)), x)
 

Reduce [F]

\[ \int \frac {(f+g x)^2 \left (A+B x+C x^2\right )}{\sqrt {d+e x} \sqrt {a+b x+c x^2}} \, dx=\int \frac {\left (g x +f \right )^{2} \left (C \,x^{2}+B x +A \right )}{\sqrt {e x +d}\, \sqrt {c \,x^{2}+b x +a}}d x \] Input:

int((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)
 

Output:

int((g*x+f)^2*(C*x^2+B*x+A)/(e*x+d)^(1/2)/(c*x^2+b*x+a)^(1/2),x)